OPENCRETIN · STAGE 3 · Λ-ITERATED TRAPPED FIELD · 2026-06-11 · REV 4 (ALL GATES GREEN INCL. R8d)
Does n(3p) float above Boltzmann?
the contested 3p-float question, answered from the converged J̄(r,ν) fixed point — not by fiat
YES — by two distinct mechanisms, both now earned by the solver.
(1) Trapped-field re-pumping: the hot-core 589 nm field photoexcites
the cool wall vapor to 1.18× Boltzmann at 0.05 atm, 1.59× at
0.01 atm — field-driven (stimulated beats thermal 5:1 at 0.01 atm),
monotonic in 1/P. (2) Chemiluminescent pumping: when the radical pool
overshoots equilibrium by Γ = (n_H·n_OH)/(n_H·n_OH)eq — the natural
state of a continuously-fed combustor — the 3p population pumps toward the
radical pool's chemical potential (ceiling: the 5.1 eV H–OH bond),
not kinetic T: core occupancy reaches 190× Boltzmann at Γ=10⁵, at 1 atm.
Only the Γ=1 corner (fully equilibrated radicals) is float-free at 1 atm —
that is v17's limit, not the device operating point. Escape factors 1.0
everywhere; g589 reported, never set; Γ is the flagged open INPUT.
The pressure axis — rev 3, all gates green
Hot mid-slice, coaxial trap R=0.97, device r_p=40 mm / gap=6 mm / H=1 m. Converged Λ-iteration (residual < 1e-8 at every P). Per-(T,P) equilibrium radical chemistry from the gibbs oracle.
P (atm)
wall n3p/Boltz
wall n̄/LTE
global g589
589 band
1.00
1.004
1.141
−4.36e-5
net absorber
0.50
1.013
1.240
−2.27e-5
net absorber
0.20
1.043
1.349
+5.83e-5
net emitter
0.10
1.091
1.419
+1.78e-4
net emitter
0.05
1.181
1.505
+3.31e-4
net emitter
0.02
1.336
1.579
+4.67e-4
net emitter
0.01
1.594
1.809
+4.77e-4
net emitter
The overshoot axis — chemiluminescent pumping, full Λ-iteration
Γ uniform on the local-equilibrium radical product [flagged crude — radical transport into the skin queued]. Reverse rate referenced to equilibrium radicals (gibbs table), per detailed balance through the H₂O reservoir.
Γ=1
10²
10³
10⁴
10⁵
1 atm — core n3p/Boltz
1.000
1.19
2.89
19.9
190
0.05 atm — core
0.996
1.08
1.80
9.0
81
0.05 atm — wall
1.18
1.19
1.26
1.95
8.8
0.05 atm — net in-band (W/m)
+2.9
+3.1
+4.6
+19.8
+170
Net in-band escape scales ~58× across the swept Γ at operating pressure — the radical pool is the brightness lever. Quench fights the pump (b ≪ Γ at 1 atm) but never pins it. Device Γ is set by fuel-feed rate vs 3-body recombination (∝ P², slow at low pressure) — kinetics model / measurement is the open INPUT.
The corrections — what the gates caught (both directions)
Rev 1 of this page reported a +30% wall float at 1 atm and a
non-monotonic pressure law. Both were artifacts of one missing term: the
chemiluminescent pump (Na + H + OH → Na(3p) + H₂O) carried no
detailed-balance reverse, and the drivers fed it 2800-K radical
fractions at the 1797-K wall — where the real equilibrium H·OH product is
~10⁶× smaller. The rate budget at the 1-atm wall ring: thermal 8.2e3 /s +
field 2.7e2 /s + spurious chem 2.4e3 /s →
reproduces the old 1.297 exactly. Honest chemistry leaves 1.004.
The same fix turned gate R8 green: the cascade now provably recovers
Saha/LTE in the dense limit (b₃ₚ = 1.0000 at n_e=10²⁴ cm⁻³, |b−1| = 2×10⁻¹⁰)
and departs at low density — the departure is earned, both signs from one
solver with no knobs. The low-pressure float and the 0.2-atm band flip
survived the correction unchanged; the 1-atm float did not.
Correction 2 — the over-correction (operator-caught). The first R8 fix
referenced the reverse rate to the actual radicals, forcing
forward/reverse = f₃ₚ identically — kinetic-T Boltzmann by construction,
silently deleting chemiluminescent pumping (the device's own mechanism: bond
energy, not kinetic temperature, sets the ceiling). Detailed balance ties rate
constants through the ~equilibrium H₂O reservoir; the reverse must be
referenced to the equilibrium radical product. Fixed, and locked with a
new gate, R8d: a chem-dominated overshoot bath must pump b → Γ (measured
b/Γ = 0.9676 vs analytic 0.967). One gate kills the flattering artifact, the
other kills the smuggled thermalization. Both directions guarded now.
Coupled core ↔ skin/corona (v3) — T is no longer prescribed
T(r) now emerges from the per-ring energy balance — chemical heating
(flagged input), net radiative exchange from the converged Λ-field (the skin is
a net absorber of the trapped D band — radiatively pumped by the core),
and radial conduction to the NaCl-wick-pinned wall. ~2.5 kW per meter of
emitter height holds the design flame (energy closure ≤0.01%). The emergent
profile validates the previously assumed parabola to ≤75 K, and the float
changes by ≤0.5% — the populations answer is robust to the thermal coupling.
At operating pressure the structure is the textbook stellar one:
subthermal core (0.996 — radiative drain beats weakened quench, a
photosphere) under a suprathermal skin (1.19 — trapped-field re-pumping,
a corona). Receipt: RECEIPT-stage3-coupled.md.
Full answers
Q. Does n(3p) float above Boltzmann at 1-atm device conditions?
Yes — if the radical pool overshoots, which a fed combustor's does. With fully equilibrated radicals (Γ=1) nothing floats at 1 atm (wall 1.004, core 0.9999 — the v17 limit). With overshoot the core pumps chemiluminescently at any pressure (2.9× at Γ=10³, 190× at Γ=10⁵, at 1 atm), bounded by the radical pool's chemical potential, not kinetic T. Separately, trapped-field re-pumping floats the wall below ~0.1 atm (1.18× @0.05, 1.59× @0.01 — stimulated beats thermal 5:1, hand-verified). The open question is no longer whether — it is the device's Γ.
Q. Is it just Boltzmann in disguise / a drive knob?
No — and this is now FULLY proven. Escape factors are 1.0, no T_exc slider, the field is solved self-consistently and g589 emerges. Gate R8 is GREEN: the same solver provably collapses to Saha/LTE in the dense limit (b₃ₚ = 1.0000 at n_e=10²⁴ cm⁻³) and departs at low density (b = 0.185 thin). Departure earned, recovery earned, one solver, no knobs.
Q. Does the float scale as 1/P, as the device intuition said?
Qualitatively yes — monotonic in 1/P (1.004 → 1.59 over 1 → 0.01 atm), saturating toward the g₁/g₀ = 3 ceiling. The onset is where the radiative+field channel out-competes quench (kQ ∝ P), around ~0.1 atm. The non-monotonic dip reported in rev 1 was the chem-pump artifact.
Q. When does the device actually radiate in-band?
Below ~0.2 atm. At 1 atm the trapped 589 band is a net absorber (g589 = −4.3e-5 — the self-reversed lid; in-band photons are recycled, the ~3% leakage + out-of-band channels carry the real exit). Below ~0.2 atm the band flips to net emitter. Low pressure is the operating regime for in-band radiative extraction.
Q. What is g589, the trapping escape factor?
Emergent, ~−4e-5 to +5e-4 depending on P — it is a geometry-and-pressure consequence, reported per ring, never the frozen 1/6000 slider it used to be. The old hand-set escape factor is retired.
Honest limits (open, not overclaimed)
819-band not transported (g819=1.0, no field) — overestimates the 3d→3p drain; the 819 escape channel (outside the trap band, a real loss) is queued.
In-plane rays only (axial transport neglected, H≫R) — underestimates trapping, so the reported float is a lower bound.
Coupled balance uses assumed k(T) and a parabolic heating shape, lumping convection; radiative cooling ledger covers only the cascade's lines (no Na₂/H₂O/CO₂ continuum), so the kW/m figures are lower bounds.
If the cited empirical kQ already includes chemical quenching by H₂O, the new DB reverse partially double-counts it (≤0.2% of down-rates at device conditions — splitting kQ is queued).
Single hot slice (j=8); full-z WGSL map queued. Wall mesh refinement near r=R queued (skin gradient steep; the outermost ring carries the float).
Doctrine held
✓ S ≠ B_λ (cascade source from occupancy, no Planck) ·
✓ populations are the state (no escape/T_exc knob; g589 emergent) ·
✓ every rate a flagged INPUT (measured / cited / assumed) ·
✓ receipt after every mutation, corrections published as revisions ·
✓ ALL GATES GREEN (R8 incl.) ·
✓ converged < 1e-8 at every pressure